Inequalities : Algebra  CAT 2016 Preparation
Question 4 the day : May 16, 2006
The question for the day is a sample practice problem in Inequalities, an Algebra Topic. The problem provides an understanding of the concept of finding out the range of values of "x" that satisfy a given inequality.
Question
Which of the following inequalities have a finite range of values of "x" satisfying them?
 x^{2} + 5x + 6 > 0
 x + 2 > 4
 9x  7 < 3x + 14
 x^{2}  4x + 3 < 0
Correct Answer  x^{2}  4x + 3 < 0. Choice (4)
Explanatory Answer
We have to find out the values of "x" that will satisfy the four inequalities given in the answer choices and check out the choice in which the range of values satisfying is finite.
Choice 1
Factorizing the given equation, we get (x + 2)(x + 3) > 0.
This inequality will hold good when both x + 2 and x + 3 are simultaneously positive or simultaneously negative.
Evaluating both the options, we get the range of values of "x" that satisfy this inequality to be x < 2 or x > 3. i.e., "x" does not lie between 2 and 3 or an infinite range of values.
Choice 2
x + 2 > 4 is a modulus function and therefore, has two options
Option 1: x + 2 > 4 or
Option 2: (x + 2) < 4.
Evaluating the two options we get the values of "x" satisfying the inequality as x > 2 and x < 6. i.e., "x" does not lie between 6 and 2 or an infinite range of values.
Choice 3
9x  7 < 3x + 14
Simplifying, we get 6x < 21 or x < 3.5. Again an infinite range of values.
Choice 4
x^2  4x + 3 < 0
Factorizing we get, (x  3)(x  1) < 0.
This inequality will hold good when one of the terms (x  3) and (x  1) is positive and the other is negative.
Evaluating both the options, we get 1 < x < 3. i.e., a finite range of values for "x".
Hence, choice 4 is the correct answer.
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